Lloyd Algorithm Research Articles

Optimal box contraction for solving linear systems via simulated and quantum annealing

Solving linear systems of equations is an important problem in engineering. Many quantum algorithms, such as the Harrow–Hassidim–Lloyd algorithm and the box algorithm, have been proposed for solving such systems. The focus of this article is on improving the efficiency of the box algorithm. The basic principle behind this algorithm is to transform the linear system into a series of quadratic unconstrained binary optimization (QUBO) problems, which are then solved on annealing machines. The computational efficiency of the box algorithm is entirely determined by the number of iterations, which, in turn, depends on the box contraction ratio, typically set to 0.5. Here, it is shown through theoretical analysis that a contraction ratio of 0.5 is sub-optimal and that a computational speed-up can be achieved with a contraction ratio of 0.2. This is confirmed through numerical experiments where a computational speed-up between 20 % to 60 % is observed when the optimal contraction ratio is used.

Solving the Hele–Shaw flow using the Harrow–Hassidim–Lloyd algorithm on superconducting devices: A study of efficiency and challenges

The development of quantum processors for practical fluid flow problems is a promising yet distant goal. Recent advances in quantum linear solvers have highlighted their potential for classical fluid dynamics. In this study, we evaluate the Harrow–Hassidim–Lloyd (HHL) quantum linear systems algorithm (QLSA) for solving the idealized Hele–Shaw flow. Our focus is on the accuracy and computational cost of the HHL solver, which we find to be sensitive to the condition number, scaling exponentially with problem size. This emphasizes the need for preconditioning to enhance the practical use of QLSAs in fluid flow applications. Moreover, we perform shots-based simulations on quantum simulators and test the HHL solver on superconducting quantum devices, where noise, large circuit depths, and gate errors limit performance. Error suppression and mitigation techniques improve accuracy, suggesting that such fluid flow problems can benchmark noise mitigation efforts. Our findings provide a foundation for future, more complex application of QLSAs in fluid flow simulations.

The HHL algorithm: Implementation and research directions

Linear systems of equations lie at the heart of numerous scientific and engineering challenges. In cutting-edge arena like artificial intelligence, machine learning and neuro-computation, these systems serve as a fundamental tool for mathematical modeling. Classical algorithms for solving linear systems have been extensively developed and forms the backbone of diverse applications across various scientific disciplines. While classical algorithms exist for solving linear systems, they often encounter limitations termed “NP-completeness” as data complexity increases. The emerging field of quantum computing offers a revolutionary approach to deal with these kinds of problems. The Harrow–Hassidim–Lloyd (HHL) algorithm tackles these challenges and opens new avenues for research. This study delves into the contemporary effectiveness of the HHL algorithm to address systems of linear equations. By examining recent research in quantum machine learning, we aim to assess the HHL algorithm’s potential to revolutionize the process of optimizing hyperparameters for machine learning models, resulting in increased efficiency and cost savings. This paper meticulously analyzes the HHL algorithm and explores its evolution from conception to the latest advancements. A comprehensive examination of the HHL algorithm, including its evolution over time, is thoroughly explored. The investigation delves into the potential challenges and limitations that might hinder the practical deployment of the HHL algorithm. Identifying these roadblocks will pave the way for future research and development efforts.

Nonconsecutive Sample Selection Based on General Lloyd Algorithm for Dual-Band Linearization

Nonconsecutive Sample Selection Based on General Lloyd Algorithm for Dual-Band Linearization

HHL algorithm with mapping function and enhanced sampling for model predictive control in microgrids

This paper presents a refined quantum Harrow Hassidim Lloyd (HHL) algorithm for microgrid control. The first novelty of the developed method is that a mapping shift function enables the original HHL algorithm to handle general linear equations with non-singular and indefinite matrix. Second, a method of Matrix Extension for Amplifying Sampling Probabilities of Intended Solution (ME-ASPI) is proposed to design the reformulated linear algebraic equations, allowing for improved sampling efficiency of the quantum tomography in the refined HHL algorithm. Then, we applied the method to solve the model predictive control (MPC) problem in nonlinear dynamical microgrids. Specifically, with the ME-ASPI method, the refined HHL algorithm can effectively obtain the intended partial optimal control inputs for MPC. The optimization of quadratic programming problem in each time step of MPC is transformed into a linear system problem, which is addressed by the proposed quantum solver through using only partial information, with the time complexity improved from O(N2.37286) classically to O(N2logN×plogp) in quantum. Numerical examples have validated the effectiveness of the refined HHL algorithm with the proposed mapping function and the ME-ASPI method. By leveraging quantum properties, the proposed method provides a hybrid quantum–classical framework for microgrid control. This generic method can also potentially tackle many other challenges in analyzing and controlling general complex engineered systems.

An algorithmic approach to convex fair partitions of convex polygons

A convex fair partition of a convex polygonal region is defined as a partition on which all regions are convex and have equal area and equal perimeter. In this article we describe an algorithm that finds such fair partition.•The Fair Partitions method finds a fair partition for any given convex polygon and any given number of regions.•Our method relies on two well-known methods: Lloyd's algorithm and the Normal Flow Algorithm.•The method proposed in this article can be used in various contexts and many real-world applications.

Cocaine Use Prediction With Tensor-Based Machine Learning on Multimodal MRI Connectome Data.

This letter considers the use of machine learning algorithms for predicting cocaine use based on magnetic resonance imaging (MRI) connectomic data. The study used functional MRI (fMRI) and diffusion MRI (dMRI) data collected from 275 individuals, which was then parcellated into 246 regions of interest (ROIs) using the Brainnetome atlas. After data preprocessing, the data sets were transformed into tensor form. We developed a tensor-based unsupervised machine learning algorithm to reduce the size of the data tensor from 275 (individuals) × 2 (fMRI and dMRI) × 246 (ROIs) × 246 (ROIs) to 275 (individuals) × 2 (fMRI and dMRI) × 6 (clusters) × 6 (clusters). This was achieved by applying the high-order Lloyd algorithm to group the ROI data into six clusters. Features were extracted from the reduced tensor and combined with demographic features (age, gender, race, and HIV status). The resulting data set was used to train a Catboost model using subsampling and nested cross-validation techniques, which achieved a prediction accuracy of 0.857 for identifying cocaine users. The model was also compared with other models, and the feature importance of the model was presented. Overall, this study highlights the potential for using tensor-based machine learning algorithms to predict cocaine use based on MRI connectomic data and presents a promising approach for identifying individuals at risk of substance abuse.

Implementation of the HHL Algorithm for Solving the Poisson Equation on Quantum Simulators

The Poisson equation is a fundamental equation of mathematical physics that describes the potential distribution in static fields. Solving the Poisson equation on a grid is computationally intensive and can be challenging for large grids. In recent years, quantum computing has emerged as a potential approach to solving the Poisson equation more efficiently. This article uses quantum algorithms, particularly the Harrow–Hassidim–Lloyd (HHL) algorithm, to solve the 2D Poisson equation. This algorithm can solve systems of equations faster than classical algorithms when the matrix A is sparse. The main idea is to use a quantum algorithm to transform the state vector encoding the solution of a system of equations into a superposition of states corresponding to the significant components of this solution. This superposition is measured to obtain the solution of the system of equations. The article also presents the materials and methods used to solve the Poisson equation using the HHL algorithm and provides a quantum circuit diagram. The results demonstrate the low error rate of the quantum algorithm when solving the Poisson equation.

Coherence and entanglement in Grover and Harrow–Hassidim–Lloyd algorithm

Coherence and entanglement in Grover and Harrow–Hassidim–Lloyd algorithm

Fusion of Centroid-Based Clustering With Graph Clustering: An Expectation-Maximization-Based Hybrid Clustering.

This article extends the expectation-maximization (EM) formulation for the Gaussian mixture model (GMM) with a novel weighted dissimilarity loss. This extension results in the fusion of two different clustering methods, namely, centroid-based clustering and graph clustering in the same framework in order to leverage their advantages. The fusion of centroid-based clustering and graph clustering results in a simple "soft" asynchronous hybrid clustering method. The proposed algorithm may start as a pure centroid-based clustering algorithm (e.g., k -means), and as the time evolves, it may eventually and gradually turn into a pure graph clustering algorithm [e.g., basic greedy asynchronous distributed interference avoidance (GADIA) (Babadi and Tarokh, 2010)] as the algorithm converges and vice versa. The "hard" version of the proposed hybrid algorithm includes the standard Hopfield neural networks (and, thus, Bruck's Ln algorithm by (Bruck, 1990) and the Ising model in statistical mechanics), Babadi and Tarokh's basic GADIA in 2010, and the standard k -means (Steinhaus, 1956), (MacQueen, 1967) [i.e., the Lloyd algorithm (Lloyd, 1957, 1982)] as its special cases. We call the "hard version" of the proposed clustering as "hybrid-nongreedy asynchronous clustering (H-NAC)." We apply the H-NAC to various clustering problems using well-known benchmark datasets. The computer simulations confirm the superior performance of the H-NAC compared to the k -means clustering, k -GADIA, spectral clustering, and a very recent clustering algorithm structured graph learning (SGL) by Kang et al. (2021), which represents one of the state-of-the-art clustering algorithms.

Two-Level Control of Multiagent Networks for Dynamic Coverage Problems.

We propose a two-level coverage control framework for a multiagent network whose members have to deploy over a given region in accordance with a (possibly time-varying) coverage density function. Our approach is based on a two-level description of the multiagent network. The first level corresponds to the probability density function of the agents' locations over a given region, in which the multiagent network is treated as one unit (macroscopic description), whereas in the second level, the network is described in terms of the collection of all individual positions of its agents (microscopic description). The goal of the multiagent network is to attain a spatial distribution that (approximately) matches the reference coverage density function (high-level coverage control problem) through local interactions of the agents of the network at the individual level (low-level coverage control problem). We address the high-level control problem by associating it with an interpolation problem in the class of Gaussian mixtures. Furthermore, we address the low-level control problem by utilizing a variation of Lloyd's algorithm with a time-varying coverage density function, which is updated at each step based on the distribution of the agents' locations. Because the high-level and the low-level coverage control problems are inherently coupled to each other, we propose an iterative scheme that combines their solutions in order to address the deployment problem in a holistic way. Finally, a set of simulation results is provided to show the effectiveness of the proposed approach.

Area Coverage Optimization using Networked Mobile Robots with State Estimation

In this paper, we present a solution to the area coverage problem using a team of mobile robots with state estimation. A group of autonomous mobile robots is deployed in a two-dimensional area of interest, for example, where communication among robots is limited or noisy as expected in a real life scenario. Each robot estimates its own state (position and orientation) using noisy range and bearing information received from other robots in its operating range. The area of interest is then divided into multiple sub-area using a voronoi tessellation. Using the classical Lloyd's algorithm, each robot employs distributed action command to move towards the centroid of the voronoi cell that it belongs yielding the maximum coverage of the area. Here we emphasize that the network of autonomous robots deployed in the environment is unknown a priori. A set of computer experiments is conducted to validate the fact that the area coverage is still possible under noisy communication among robots deployed in a two-dimensional area.

Computer Vision Aided Codebook Design for MIMO Communications Systems

MmWave communications systems usually rely on analog or hybrid analog/digital architectures and thus need a predefined codebook to perform beamforming. Traditional codebooks are designed for universal environments, although in practice a particular BS will only serve a particular environment. In this paper, we propose novel site-specific codebook design methods by utilizing the visual information captured through cameras. Different from other site-specific codebook design methods that require a large amount of measured channel state information (CSI), the proposed ones need only a simple snapshot of the environment followed by efficient computer vision (CV) techniques. Thus the proposed CV-aided codebook design reduces the overhead of communications system, such as the cost of time, human resources, as well as the hardware installation and calibration. Specifically, we propose a CV-based approach that detects the LOS area around the BS and reconstructs the LOS channel vectors set (CVS).With this knowledge, we build a vision-based beam codebook using Lloyd algorithm. Further, we design a FusionNet to generate the codebook that can serve the non-line-of-sight (NLOS) users. The simulation results demonstrate the effectiveness of the proposed CV-aided codebook design methods and their superiority compared to the conventional methods.

An Effective and Efficient Algorithm for K-Means Clustering With New Formulation

K-means is one of the most simple and popular clustering algorithms, which implemented as a standard clustering method in most of machine learning researches. The goal of K-means clustering is finding a set of cluster centers and minimizing the sum of squared distances between each sample and its nearest clustering center. In this paper, we proposed a novel K-means clustering algorithm, which reformulate the classical K-Means objective function as a trace maximization problem and then replace it with a new formulation. The proposed algorithm does not need to calculate the cluster centers in each iteration and requires fewer additional intermediate variables during the optimization process. In addition, we proposed an efficient iterative re-weighted algorithm to solve the involved optimization problem and provided the corresponding convergence analysis. The proposed algorithm keeps a consistent computational complexity as Lloyd's algorithm, <inline-formula><tex-math notation="LaTeX">$\mathcal {O}(ndk)$</tex-math></inline-formula> , but shows a faster convergence rate in experiments. Extensive experimental results on real world benchmark datasets show the effectiveness and efficiency of the proposed algorithm.

Carleman linearization approach for chemical kinetics integration toward quantum computation

The Harrow, Hassidim, Lloyd (HHL) algorithm, known as the pioneering algorithm for solving linear equations in quantum computers, is expected to accelerate solving large-scale linear ordinary differential equations (ODEs). To efficiently combine classical and quantum computers for high-cost chemical problems, non-linear ODEs (e.g., chemical reactions) must be linearized to the highest possible accuracy. However, the linearization approach has not been fully established yet. In this study, Carleman linearization was examined to transform nonlinear first-order ODEs of chemical reactions into linear ODEs. Although this linearization theoretically requires the generation of an infinite matrix, the original nonlinear equations can be reconstructed. For the practical use, the linearized system should be truncated with finite size and the extent of the truncation determines analysis precision. Matrix should be sufficiently large so that the precision is satisfied because quantum computers can treat such huge matrix. Our method was applied to a one-variable nonlinear dot{y}=-{y}^{2} system to investigate the effect of truncation orders and time step sizes on the computational error. Subsequently, two zero-dimensional homogeneous ignition problems for H2–air and CH4–air gas mixtures were solved. The results revealed that the proposed method could accurately reproduce reference data. Furthermore, an increase in the truncation order improved accuracy with large time-step sizes. Thus, our approach can provide accurate numerical simulations rapidly for complex combustion systems.

Distributed Control for Human-Swarm Interaction In Non-Convex Environments using Gaussian Mixture Models

This paper presents a novel implementation of a human-swarm interface that allows humans to define an area with desired shape to be reached by a multi-robot system. Human-swarm interaction can be useful in order to exploit human intelligence and knowledge for the operation of swarm robots. The proposed work deals with limitations usually met when dealing with real-world implementation, e.g. limited sensing capabilities of the agents and hard conditions where communication is difficult or even completely denied. Gaussian Mixture Models are exploited in order to define an appropriate probability density function of the environment based on the area selected by a human operator. Then, velocity input for each robot is calculated in a distributed manner using Voronoi tessellation and Lloyd's algorithm. Finally, results of both virtual and real-world tests are presented, showing the final configuration reached by the multi-robot system in comparison with the desired region defined on the graphical interface.

Quantum Fuzzy Regression Model for Uncertain Environment

In the era of big data, traditional regression models cannot deal with uncertain big data efficiently and accurately. In order to make up for this deficiency, this paper proposes a quantum fuzzy regression model, which uses fuzzy theory to describe the uncertainty in big data sets and uses quantum computing to exponentially improve the efficiency of data set preprocessing and parameter estimation. In this paper, data envelopment analysis (DEA) is used to calculate the degree of importance of each data point. Meanwhile, Harrow, Hassidim and Lloyd (HHL) algorithm and quantum swap circuits are used to improve the efficiency of high-dimensional data matrix calculation. The application of the quantum fuzzy regression model to small-scale financial data proves that its accuracy is greatly improved compared with the quantum regression model. Moreover, due to the introduction of quantum computing, the speed of dealing with high-dimensional data matrix has an exponential improvement compared with the fuzzy regression model. The quantum fuzzy regression model proposed in this paper combines the advantages of fuzzy theory and quantum computing which can efficiently calculate high-dimensional data matrix and complete parameter estimation using quantum computing while retaining the uncertainty in big data. Thus, it is a new model for efficient and accurate big data processing in uncertain environments.

Cluster analysis based on multidimensional means based by results of All-Russian Agricultural Census of 2016 using Python

The article describes the stages and results of the multidimensional classification of personal subsidiary plots (PSP) according to the All-Russian Agricultural Census of 2016 (VSHP-2016). This classification was carried out on the basis of four prepared sets of initial data, two of which are synthetic indicators in the form of multidimensional means, and implemented using the Python programming language, including the skit-learn library for clustering and the matplotlib library for graphical visualization of the resulting split of households into homogeneous groups. The original data is normalized using the L2 Normalization method, also known as Spatial Sign Preprocessing. Cluster analysis is carried out using the k-means method based on the Lloyd algorithm, the number of clusters is determined using the Silhouette Coefficient. The results are visualized through bar charts, 2D and 3D scatter plots.

Exact Clustering in Tensor Block Model: Statistical Optimality and Computational Limit

AbstractHigh-order clustering aims to identify heterogeneous substructures in multiway datasets that arise commonly in neuroimaging, genomics, social network studies, etc. The non-convex and discontinuous nature of this problem pose significant challenges in both statistics and computation. In this paper, we propose a tensor block model and the computationally efficient methods, high-order Lloyd algorithm (HLloyd), and high-order spectral clustering (HSC), for high-order clustering. The convergence guarantees and statistical optimality are established for the proposed procedure under a mild sub-Gaussian noise assumption. Under the Gaussian tensor block model, we completely characterise the statistical-computational trade-off for achieving high-order exact clustering based on three different signal-to-noise ratio regimes. The analysis relies on new techniques of high-order spectral perturbation analysis and a ‘singular-value-gap-free’ error bound in tensor estimation, which are substantially different from the matrix spectral analyses in the literature. Finally, we show the merits of the proposed procedures via extensive experiments on both synthetic and real datasets.

Improved circuit implementation of the HHL algorithm and its simulations on QISKIT

In 2019, Yonghae Lee et al. combined the circuit implementation of the Harrow–Hassidim–Lloyd (HHL) algorithm with a classical computer, and designed a hybrid HHL algorithm to reduce experimental errors caused by decoherence and so on. However, the improvement is achieved only in the auxiliary quantum coding phase, and no quantum resource reduction is done on the quantum phase estimation and inverse quantum phase estimation stages. At the same time, the circuit improvement illustration on a 2times 2 linear system just has the result and no specific process. In this paper, based on the idea of the hybrid HHL algorithm and a generic circuit of HHL algorithm, an improved circuit implementation of the HHL algorithm is proposed. The feasibility of the improved circuit implementation of the HHL algorithm is verified by IBM's qiskit. The improved circuit illustrations on a 4times 4 linear system show that the improved circuit implementation of the HHL algorithm can effectively reduce quantum resources without losing the fidelity of the results. Thus the improved circuit implementation of the HHL algorithm can further avoid some result errors than the existing implementation methods.